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Chemistry and Physics of Lipids 164 (2011) 177–183

Contents lists available at ScienceDirect

Chemistry and Physics of Lipids

journa l homepage: www.e lsev ier .com/ locate /chemphys l ip

eview

ivotal surfaces in inverse hexagonal and cubic phases of phospholipids andlycolipids

erek Marsh ∗

ax-Planck-Institut für biophysikalische Chemie, Am Fassberg 11, 37077 Göttingen, Germany

r t i c l e i n f o

rticle history:eceived 26 November 2010ccepted 21 December 2010

a b s t r a c t

Data on the location and dimensions of the pivotal surfaces in inverse hexagonal (HII) and inverse cubic(QII) phases of phospholipids and glycolipids are reviewed. This includes the HII phases of dioleoylphosphatidylethanolamine, 2:1 mol/mol mixtures of saturated fatty acids with the corresponding diacyl

vailable online 6 January 2011

eywords:on-Lamellar phase

nverse hexagonal phasenverse cubic phase

phosphatidylcholine, and glucosyl didodecylglycerol, and also the QII230/G gyroid inverse cubic phases of

monooleoylglycerol and glucosyl didodecylglycerol. Data from the inverse cubic phases are largely com-patible with those from inverse hexagonal HII-phases. The pivotal plane is located in the hydrophobicregion, relatively close to the polar–apolar interface. The area per lipid at the pivotal plane is similar insize to lipid cross-sectional areas found in the fluid lamellar phase (L�) of lipid bilayers.

pontaneous curvatureivotal plane

© 2011 Elsevier Ireland Ltd. All rights reserved.

ontents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1772. Mathematical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

2.1. Pivotal surfaces in inverse hexagonal (HII) phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1782.2. Pivotal surfaces in inverse cubic (QII) phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

3. Inverse hexagonal (HII) phases of phospholipids and glycolipids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1793.1. Phosphatidylethanolamines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1793.2. Lipid mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1803.3. Glycosyl dialkylglycerols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

4. Inverse cubic (QII) phases of monoglycerides, phospholipids and glycolipids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1814.1. Monoacylglycerols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

4.2. Lipid mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

. . .. . . .. . .. . . .

o o p

t the pivotal plane in inverse hexagonal (HII) phases (Leikin et al.,996). The pivotal plane is the surface at which the area remainsonstant as the curvature in the HII phase is changed by varying theater content (i.e., ∂Ap/∂c̄ = 0). This definition is restricted to this

∗ Corresponding author. Tel.: +49 5512011285; fax: +49 5512011501.E-mail address: dmarsh@gwdg.de

009-3084/$ – see front matter © 2011 Elsevier Ireland Ltd. All rights reserved.oi:10.1016/j.chemphyslip.2010.12.010

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

type of experiment and does not correspond to the neutral surface,which is the surface where bending and stretching contributions tothe elastic energy are uncoupled, and is a quite general property ofthe lipid–water system (see Kozlov and Winterhalter, 1991; Leikinet al., 1996). An equivalent definition of pivotal surfaces appliesto bicontinuous inverse cubic (QII) phases that are based on infi-nite periodic minimal surfaces (IPMS) (Templer, 1995). In this case,the parallel pivotal surfaces of the lipid bilayer are symmetricallydisplaced from the IPMS.

4.3. Glycosyl dialkylglycerols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction

Lipids with an intrinsic tendency to form nonlamellar phases inaqueous dispersion are characterized by a high spontaneous cur-vature, c = 1/R . The latter is defined by the radius of curvature, R ,

Pivotal surfaces therefore assume a central role in characterizingthe tendency of lipid membranes to form nonlamellar phases, andcorrespondingly for the functional consequences of curvature frus-tration on membrane-embedded enzymes and transport systems

178 D. Marsh / Chemistry and Physics o

Fig. 1. Position of the pivotal plane (heavy broken lines), relative to the lipid–waterips

(idc

2

2

ttiov(ftral

t

R

wmrga

A

St

Ap = 2vl

a�l

[�o + 2��

(lpa

)2]

(7)

Table 1Euler characteristic, �, and dimensionless surface area, �o , per unit cell for bicon-tinuous inverse cubic phases based on infinite periodic minimal surfaces.

nterface, in inverse hexagonal (HII) phases. Ap is the area per lipid at the pivotallane and vp is the volume per lipid (of total volume vl) that lies outside the pivotalurface. Rp is the radius of the pivotal surface.

see, e.g., Marsh, 2007). The purpose of the present communications to catalogue and compare the pivotal surfaces for the availableatabase of phospholipid and glycolipid inverse hexagonal andubic phases.

. Mathematical background

.1. Pivotal surfaces in inverse hexagonal (HII) phases

Removal of water from an inverse nonlamellar phase increaseshe surface curvature of the lipid component. The pivotal surface ishat plane which maintains constant area as the curvature changesn response to osmotic stress or decreasing water content. The piv-tal surface is offset from the lipid–water interface and encloses aolume per lipid, vp, that is smaller than the total lipid volume, vl

see Fig. 1). In terms of conventional derivations of the dimensionsor inverse phases, which are referred to the lipid–water interface,hose for the pivotal surface are obtained similarly by using theeduced lipid volume, vp, instead of the total lipid volume, vl , andlso the reduced lipid volume fraction �l(vp/vl) in place of the totalipid volume fraction, �l.

The radius of curvature of the pivotal surface in HII-phases ishus given by:

p = aH

√√3

2�

(1 − vp

vl�l

)(1)

here aH is the hexagonal lattice constant. This expression is deter-ined solely by geometry and applies to any dividing surface. The

adius of the lipid–water interface in HII-phases, for instance, isiven by substituting vp = vl in Eq. (1) (cf. Seddon et al., 1984). Therea per lipid at the pivotal surface is correspondingly given by:

4�Rpvl

p = √

3a2H�l

(2)

ubstituting for Rp in Eq. (2) then gives the dependence ofhe hexagonal lattice constant on water content, �w = 1 − �l (see

f Lipids 164 (2011) 177–183

Templer et al., 1998b):

aH = 2vl

Ap�l

√2�√

3

(1 − vp

vl�l

)(3)

Hence Ap and vp, which are independent of water content, may bedetermined from measurements of the hexagonal lattice constant,aH, in a swelling experiment. Success in fitting aH as a function of�l according to the above expression demonstrates the existenceof a well-defined pivotal plane (which must not necessarily be thecase for high curvatures – see Leikin et al., 1996).

Having determined the position of the pivotal plane, the radiusof curvature of the pivotal surface, Rp, then can be determinedfrom Eq. (1) by using the value of the pivotal lipid volume vp

obtained from the swelling experiments. Eq. (3) can have numericaladvantages over the original linearised form introduced in Leikinet al. (1996), in that the parameters of the pivotal surface may beobtained with better precision (cf. Di Gregorio and Mariani, 2005).

2.2. Pivotal surfaces in inverse cubic (QII) phases

Bicontinuous inverse cubic phases in which both lipid and watercomponents are continuous and constitute a bilayer motif, arebased on periodic minimal surfaces. The IPMS defines the midplaneof the lipid bilayer, and the pivotal surfaces of each monolayer aredisplaced from it at a distance lp.

Minimal surfaces are saddle shaped and the mean curvature iszero everywhere on the surface (c1 = −c2, and c̄ = 0). Thus, for eachcubic unit cell, the surface area at the pivotal surface is:

Acell(lp) = Acell(0)(

1 + l2p〈c̄2G〉

)(4)

where Acell(0) is the area of the minimal surface, per cubic unit cell,and 〈c̄2

G〉 is the mean Gaussian curvature of the minimal surface (see,e.g., Marsh, 2006). The mean Gaussian curvature is given from theGauss–Bonnet theorem by:

〈c̄2G〉 = 2��

Acell(0)(5)

where � is the Euler characteristic of the surface, per cubic unit cell.Hence the surface area per unit cell of one lipid monolayer is givenby (Anderson et al., 1988):

Acell(lp) = �oa2 + 2��l2p (6)

where a is the cubic lattice constant and �o = Acell(0)/a2 is thedimensionless area per cubic unit cell of the minimal surface. Val-ues of � and �o for the common cubic phases that are based on anIPMS are listed in Table 1.

The area per lipid molecule at the pivotal surface, Ap =2Acell(lp)vl/(a3�l), is hence given by:

Cubic phase Minimal surface � �o

QII224 (Pn3m) D −2 1.9189

QII229 (Im3m) P −4 2.3451

QII230 (Ia3d) G −8 3.0915

ysics of Lipids 164 (2011) 177–183 179

Ts

V

T(

ts

T�nbd

(s

a

w(fef

3g

3

tienT

�

wevF1A

ctaiA

0

2

4

6

8

403020100

0.2

0.3

0.4

0.5

0.6

0.7

dis

tance,

aH,

2R

p,

2R

w

(nm

)

2Rp

2Rw

aH

HII

are

a/lip

id,

Ap,

Aw (

nm

2)

water/lipid, nw (mol/mol)

Ap

Aw

(18:1cΔ9)

2PE, 22

oC

Fig. 2. Dependence on water/lipid ratio, nw , of: (upper panel) the hexagonal latticeconstant, aH , the cylinder radii of the pivotal surface (Rp) and lipid–water interface

D. Marsh / Chemistry and Ph

he volume per unit cell that is contained between the minimalurface and the pivotal surface is given by:

cell(lp) =lp∫0

Acell(z) dz = Acell(0)lp(

1 + 13

l2p〈c̄2G〉

)(8)

hus the volume fraction between pivotal and minimal surfaces,vp/vl)�l = 2Vcell(lp)/a3, is given by (Turner et al., 1992):

vp

vl�l = 2�o

lpa

+ 4�

3�

(lpa

)3

(9)

Eq. (9) is a cubic equation that has two positive roots (�l < 1),he physically realistic of which gives the following solution for theeparation of the pivotal surfaces in bicontinuous cubic phases:

dp = 2lp = a

√8�o

�|�| cos

(� + ϑp

3

),

where cos ϑp = �lvp

vl

√9�|�|8�3

o

(10)

his is determined solely by the water content of the phase,w = 1 − �l, and the cubic lattice constant, a. The lipid bilayer thick-ess, dl, and the area per lipid at the lipid–water interface, Aw, cane obtained in a similar way by using the full lipid length, ll, that iseduced from Eq. (10) with vp = vl .

Substituting for lp/a from Eq. (10) in the expression for Ap (Eq.7)) then gives the following dependence of the cubic lattice con-tant on the water (or lipid) content of the cubic phase:

= 2�ovl

Ap�l

[1 − 4 cos2

(� + ϑp

3

)](11)

here ϑp depends on �l and vp according to the second part of Eq.10). As in the case of HII-phases, Ap and vp can thus be determinedrom the dependence of the lattice constant a on �l in a swellingxperiment. Values for the separation, dp, of the pivotal planes thenollow by using these determinations of vp.

. Inverse hexagonal (HII) phases of phospholipids andlycolipids

.1. Phosphatidylethanolamines

The dependence of the hexagonal lattice constant on water con-ent of the dioleoyl phosphatidylethanolamine HII-phase at 22 ◦Cs shown in the upper panel of Fig. 2 (Rand et al., 1990; Grunert al., 1986). Up to the maximum hydration at a water/lipid ratio ofw = 19 mol/mol, the dependence of aH on nw is well fit by Eq. (3).he lipid volume fraction is:

l = 11 + (vw/vl)nw

(12)

here vw is the volume of a water molecule. The nonlin-ar least-squares fit yields the following optimized parameters:p/vl = 0.604 ± 0.024 and Ap/vl = 0.578 ± 0.011 nm−1 at 22 ◦C.or a molecular volume of vl = 1.215 nm3 (Tate and Gruner,989), this yields an area per lipid at the pivotal plane of:p = 0.703 ± 0.014 nm2.

The lower panel of Fig. 2 contrasts the dependence on water

ontent of the area per lipid at the pivotal plane (Ap) with that athe lipid/water interface (Aw). Ap remains constant, as it must forwell-defined pivotal plane, whereas Aw increases steeply with

ncreasing water content up to the point of maximum hydration.p is greater than Aw, and correspondingly for the radii Rp and Rw of

(Rw), and (lower panel) the areas per lipid at the pivotal surface (Ap) and lipid–waterinterface (Aw), for the inverse hexagonal (HII) phase of 1,2-dioleoyl-sn-glycero-3-phosphoethanolamine at 22 ◦C. Data from Rand et al. (1990); Gruner et al. (1986).The solid line for aH at nw ≤ 19 is a nonlinear, least-squares fit of Eq. (3) (with Eq.(12)) for the pivotal plane.

the pivotal plane and lipid–water interface, respectively, indicatingthat the pivotal plane lies inside the lipid region (because vp < vl).

At excess water, the radius of the pivotal plane in the HII-phase isthe spontaneous radius of curvature, Ro,p, that specifies the intrinsicpropensity of the lipid assembly to bend (see, e.g., Marsh, 1996,2006). For lipids with low intrinsic curvature that do not form HIIphases spontaneously, these can be induced by addition of excessliquid alkane, which alleviates the chain packing frustration that isinherent to the hexagonal phase geometry (Gruner, 1985).

Table 2 gives the dimensions of the HII-phase for phos-phatidylethanolamines, including those of the pivotal plane. Thetemperature dependence for dioleoyl phosphatidylethanolamine,(18:1c�9)2PE, is deduced from the data of Tate and Gruner (1989),with additional data at 22 ◦C from Rand et al. (1990) and Randand Fuller (1994). An aggregate pivotal volume of vp ≈ 0.79 nm3

is obtained for (18:1c�9)2PE at 20–25 ◦C. Taking a volume perCH2 group of 0.027 nm3 and ratios of volumes CH3/CH2 = 2 andCH/CH2 = 0.85 (Marsh, 2010), the pivotal plane is predicted to liein the region of the C4–C5 atom of the chains. A similar calcula-tion for (20:0)2PE at 99 ◦C, for which the pivotal volume is vp =1.18 ± 0.05 nm3, puts the pivotal plane close to the chain carbonyls.For the shorter ether-linked (O-12:0)2PE at 135 ◦C, on the other

3

hand, the pivotal volume is predicted to be vp = 0.11 ± 0.09 nmand the volume per CH2 group is 0.0286 nm3, which implies thatthe pivotal plane is situated close to the terminal methyl groups.This latter is an unphysical result, which arises because packingfrustration becomes dominant relative to curvature energy in the

180 D. Marsh / Chemistry and Physics of Lipids 164 (2011) 177–183

Table 2Dimensions at the pivotal plane and lipid–water interface for HII-phases of phosphatidylethanolamines in excess water.

Lipid T (◦C) nwmax (mol/mol) Ro,p (nm) Rw (nm) Ap

a (nm2) Aw (nm2) vp/vla Ref.b

(18:1c�9)2PE 10 18.0 3.78 2.28 0.785 ± 0.029 0.473 0.215 ± 0.086 115 17.5 3.42 2.21 0.732 ± 0.017 0.474 0.399 ± 0.044 120 16.9 2.91 2.15 0.639 ± 0.016 0.471 0.650 ± 0.034 120 19.5 3.06 2.26 0.698 ± 0.003 0.506 0.616 ± 0.005 222 19.1 3.00 2.21 0.703 ± 0.014 0.518 0.604 ± 0.024 3,422 20.3 3.10 2.26 0.729 ± 0.013 0.539 0.562 ± 0.027 422 18.1 2.89 2.16 0.663 ± 0.008 0.502 0.649 ± 0.015 525 16.6 2.84 2.10 0.643 ± 0.016 0.475 0.659 ± 0.033 130 16.2 2.78 2.05 0.646 ± 0.020 0.476 0.665 ± 0.040 135 16.0 2.73 2.00 0.655 ± 0.021 0.480 0.663 ± 0.041 140 15.6 2.62 1.95 0.642 ± 0.019 0.479 0.697 ± 0.037 145 15.2 2.53 1.91 0.634 ± 0.019 0.478 0.719 ± 0.035 150 14.9 2.45 1.86 0.630 ± 0.021 0.480 0.738 ± 0.035 155 14.7 2.42 1.83 0.637 ± 0.022 0.483 0.735 ± 0.037 160 14.5 2.30 1.79 0.645 ± 0.024 0.485 0.774 ± 0.039 165 14.2 2.28 1.75 0.633 ± 0.025 0.486 0.763 ± 0.039 170 14.0 2.22 1.72 0.630 ± 0.020 0.488 0.776 ± 0.031 175 13.8 2.17 1.68 0.633 ± 0.018 0.491 0.782 ± 0.028 180 13.7 2.16 1.66 0.645 ± 0.018 0.495 0.773 ± 0.032 185 13.7 2.13 1.64 0.650 ± 0.020 0.500 0.777 ± 0.029 190 13.6 2.13 1.62 0.663 ± 0.029 0.505 0.767 ± 0.043 1

(20:0)2PE 99 15.7 2.53 2.11 0.557 ± 0.033 0.464 0.845 ± 0.038 6(O-12:0)2PE 135 15.8 (2.64)c 1.60 (1.04 ± 0.04)c 0.632 (0.114 ± 0.09)c 6

uner et al. (1986); 4. Rand et al. (1990); 5. Rand and Fuller (1994); 6. Seddon et al. (1984).otal plane inaccurate (see text).

hwpw

s(bippoacttdctlpsr(nb

3

dtcniaetp

0.2

0.4

0.6

0.8

1.0

1.2

1.4

908070605040302010

1.5

2.0

2.5

3.0

3.5

4.0

0.4

0.5

0.6

0.7

0.8

vo

lum

e, v (

nm

3)

vp

vl

rad

ius, R

(n

m)

temperature (oC)

Rp

Rw

are

a, A

(n

m2) A

p

Aw

Fig. 3. Dependence on temperature of the properties of the pivotal surface(solid symbols) and lipid–water interface (open symbols) for the HII-phase of

9

a Deduced from the water dependence.b References: 1. Tate and Gruner (1989); 2. Di Gregorio and Mariani (2005); 3. Grc Packing stresses dominate over bending elasticity, making estimation of the piv

igh-temperature HII-phases of this short-chain lipid. Comparisonith the result for long-chain (20:0)2PE suggests, however, that theivotal plane moves outwards, deeper into the hydrocarbon region,ith decreasing chain length.

Fig. 3 compares the temperature dependences of the dimen-ions of the pivotal plane (solid symbols) for the HII-phase of18:1c�9)2PE with those for the lipid–water interface (open sym-ols). The area per lipid (Aw) and radius (Rw) at the lipid–water

nterface, and the total lipid volume (vl), vary smoothly with tem-erature (see also Tate and Gruner, 1989). In contrast, the areaer lipid (Ap) and radius (Rp) of the pivotal plane, and the piv-tal lipid volume (vp), change rapidly between 10 and 20 ◦C beforessuming a less steep temperature dependence. These sharp initialhanges are probably a feature of the low-temperature region closeo 0 ◦C, where the lipid chains are stiffer and the packing stresseshat arise from the nonuniform radial geometry of the HII-phaseominate over the bending stresses. At higher temperatures, thehains are more flexible, packing stresses are less important, andhe free energy is dominated by bending elasticity. The data in theower panel of Fig. 3 are of particular interest, because the tem-erature dependence of the spontaneous radius of curvature (Ro,p,olid triangles) of the pivotal surface for (18:1c�9)2PE has not beeneported previously, only that of the lipid–water interface (Ro,w)Tate and Gruner, 1989). The pressure dependence of the sponta-eous curvature at the pivotal plane is reported for (18:1c�9)2PEy Pisani et al. (2003).

.2. Lipid mixtures

Table 3 gives the dependence on lipid chain length of theimensions of the HII-phases that are formed by 1:2 mol/mol mix-ures of disaturated phosphatidylcholines with fatty acids of theorresponding chain lengths. The fatty acids are in the proto-ated (uncharged) state and thus have a strong tendency to form

nverted phases on admixture with phosphatidylcholines (Marshnd Seddon, 1982; Rama Krishna and Marsh, 1990; Heimburgt al., 1990). Data are given at a temperature that is 10 ◦ abovehe transition to the HII-phase for the respective chain lengths. Thearameters of the pivotal planes for these mixtures were presented

(18:1c� )2PE. Top panel: area per lipid at the pivotal surface (Ap , solid circles), andat the lipid–water interface (Aw , open circles). Middle panel: pivotal lipid volume(vp , solid squares), and total lipid volume (vl , open squares). Bottom panel: radius ofthe pivotal surface (Rp , solid triangles), and of the lipid–water interface (Rw , opentriangles). Primary data from Tate and Gruner (1989).

D. Marsh / Chemistry and Physics of Lipids 164 (2011) 177–183 181

Table 3Dimensions at the pivotal plane and lipid–water interface for HII phases of phosphatidylcholine:fatty acid 1:2 mol/mol mixtures with matched saturated chains in excesswater.

Lipid T (◦C) nwmax a (mol/mol) Ro,p (nm) Rw (nm) Ap

a,b (nm2) Awa (nm2) vp/vl

b Ref.c

(12:0)2PC/(12:0)FA 1:2 mol/mol 40 10.5 2.14 1.56 0.553 ± 0.015 0.403 0.681 ± 0.029 1(14:0)2PC/(14:0)FA 1:2 mol/mol 60 9.6 2.10 1.48 0.562 ± 0.015 0.474 0.693 ± 0.023 1(16:0)2PC/(16:0)FA 1:2 mol/mol 70 14.1 2.39 1.93 0.546 ± 0.026 0.441 0.789 ± 0.035 2(18:0)2PC/(18:0)FA 1:2 mol/mol 80 13.8 2.29 1.82 0.574 ± 0.027 0.456 0.794 ± 0.033 2(20:0)2PC/(20:0)FA 1:2 mol/mol 90 13.5 2.30 1.78 0.586 ± 0.029 0.453 0.795 ± 0.033 2

N

itc

tfiAldbos

3

wgdhpitTg0t

F(of(

gives an area per lipid of: Ap = 0.361 ± 0.007 nm2 at the pivotal planeat 25 ◦C. Table 5 gives the temperature dependence of the dimen-sions for the gyroid QII

230/G cubic phase of (18:1c�9/0:0)MG thatis deduced from the dependence on water content.

3

4

5

6

, 2R

p,

2R

w (

nm

)

aH

Rp

Rw

ote: the fatty acid is protonated.a Value per two chains.b Deduced from water dependence.c References: 1. Templer et al. (1998b); 2. Seddon et al. (1997).

n the original study by Templer et al. (1998b). The present evalua-ions, which are mean values per two chains instead of for the 1:2omplex, agree with those values.

It was found by Templer et al. (1998b) that the location ofhe pivotal plane and its area does not change with chain lengthor the 1:2 mol/mol phosphatidylcholine–fatty acid mixtures. Thiss illustrated in Fig. 4, which shows the dependence of vp andp on lipid chain length, nC. The difference between vl , the total

ipid molecular volume, and vp remains essentially constant, asoes the value of Ap, for the different chain lengths. Estimatesased on a methylene group volume of 0.027 nm3 place the piv-tal plane in the region of the C2 atom of the chains for eachystem.

.3. Glycosyl dialkylglycerols

Fig. 5 shows the dependence of the HII-phase dimensions onater content for a glycolipid system, monoglucosyl didodecyl-

lycerol (rac-(O-12:0)2Glc�DG). This is exactly comparable to theata for dioleoyl phosphatidylethanolamine in Fig. 2. The maximumydration (nw ≈ 12 mol/mol) is less for the glycolipid than for thehospholipid, and the dimensions of the HII-phase are correspond-

ngly smaller. Table 4 gives the HII-phase dimensions, includinghose of the pivotal plane, for three temperatures in the HII-regime.

he volume of a CH2 group in the HII phase of glucosyl dialkyl-lycerols is 0.0265 nm3 (Marsh, 2010) and the pivotal volume is.586 nm3 at 80 ◦C. This places the pivotal plane in the region ofhe C2–C3 atom of the alkyl chains.

20181614120.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

piv

ota

l lip

id a

rea

, A

p (

nm

2 )

piv

ota

l (v

p)

or

lipid

(v

l) volu

me

(n

m3)

chain length, nC (C-atoms)

(nC:0)2PC/(nC:0)FA 1:2 mol/mol

HII-phase

Ap

vl

vp

ig. 4. Chain-length dependence of the pivotal volume (vp , circles) and surface areaAp , triangles), per two lipid chains, in the HII-phase from 1:2 mol/mol mixturesf saturated diacyl phosphatidylcholines with the corresponding fatty acid. Datarom Table 3. The lipid molecular volume per two chains, vl , is given by the squaresTempler et al., 1998b). Sloping lines are linear regressions.

4. Inverse cubic (QII) phases of monoglycerides,phospholipids and glycolipids

4.1. Monoacylglycerols

Fig. 6 shows the dependence on water content, nw, of the cubiclattice constants, a, for the QII

230/G (Ia3d) gyroid inverse cubicphase of monooleoylglycerol, (18:1c�9/0:0)MG, at 25 ◦C (Chungand Caffrey, 1994). The dependence is reasonably well fit by Eq.(11), yielding the optimized parameters: vp/vl = 0.582 ± 0.062 andAp/vl = 0.574 ± 0.011 nm−1 for the pivotal surface. With a molecu-lar volume of vl = 0.6285 nm3 (based on the density at 20 ◦C), this

0

1

2

35302520151050

0.2

0.3

0.4

0.5

0.6

0.7

rac-(O-12:0)2GlcβDG, 75

oC

dis

tan

ce

, a

H

HII

are

a/lip

id,

Ap,

Aw (

nm

2)

water/lipid, nw (mol/mol)

Ap

Aw

Fig. 5. Dependence on water/lipid ratio, nw , of: (upper panel) the hexagonal lat-tice constant, aH , the cylinder radii of the pivotal surface (Rp) and lipid–waterinterface (Rw), and (lower panel) the areas per lipid at the pivotal surface (Ap) andlipid–water interface (Aw), for the inverse hexagonal (HII) phase of rac-didodecyl-�-d-glucosylglycerol at 75 ◦C. Data from Turner et al. (1992). The solid line for aH atnw ≤ 12 is a nonlinear, least-squares fit of Eq. (3) (with Eq. (12)) for the pivotal plane.

182 D. Marsh / Chemistry and Physics of Lipids 164 (2011) 177–183

Table 4Dimensions of the pivotal plane and lipid–water interface for the HII-phase of �-d-glucosyl didodecyl-rac-glycerol in excess watera.

T (◦C) nwmax (mol/mol) Ro,p (nm) Rw (nm) Ap

b (nm2) Aw (nm2) vp/vlb

75 12.1 2.24 1.55 0.674 ± 0.016 0.467 0.607 ± 0.02980 11.9 2.24 1.51 0.701 ± 0.034 0.471 0.582 ± 0.06085 12.0 2.23 1.49

a Data from Turner et al. (1992).b Deduced from water dependence.

109876

12.5

13.0

13.5

14.0

14.5

15.0

cubic

lattic

e c

onsta

nt,

a (

nm

)

water/lipid, nw (mol/mol)

Q230/GII

, Ia3d

(18:1cΔ9)MG, 25

oC

F12o

poh2sfo

4

fQcTtos

TDi

N(

(

umes are larger than those in the HII-phase (cf. Table 4) and, inconsequence, the pivotal plane is predicted to lie within the regionof the polar–apolar interface.

14

)

rac-(O-12:0)2GlcβDG, 45

oC

ig. 6. Dependence of the cubic lattice parameter, a, on water/lipid ratio, nw , for-oleoyl-sn-glycerol, (18:1c�9/0:0)MG, in the QII

230/G (Ia3d) gyroid cubic phase at5 ◦C. Data from Chung and Caffrey (1994). Solid line is a nonlinear, least-squares fitf Eq. (11) (with Eq. (12)) for the pivotal plane relative to the IPMS.

Note that, although the QII224/D (Pn3m) diamond inverse cubic

hase occurs in (18:1c�9/0:0)MG, a realistic location for the piv-tal surface has been found only when this phase is stabilized byigh concentrations of saccharides (Pisani et al., 2001; Saturni et al.,001). Presumably, in the absence of sugars, packing stresses and/ortretching elasticity are more important than curvature elasticityor the stability of the QII

224/D (Pn3m) structure (see also discussionf (O-12:0)2PE in Section 3.1).

.2. Lipid mixtures

Table 6 gives the dimensions for the pivotal surfaces of the dif-erent inverse cubic phases, QII

230/G (Ia3d), QII229/P (Im3m) and

II224/D (Pn3m), that are formed by a dilauroyl phosphatidyl-

holine:lauric acid 1:2 mol/mol mixture (Templer et al., 1998b).

hese values are determined from the dependence of the cubic lat-ice parameters on water content (Eq. (11)). As was assumed in theriginal work (Templer et al., 1998b), the parameters of the pivotalurfaces are found to be similar to those for the inverse hexago-

able 5imensions of the pivotal plane and lipid–water interface for the QII

230/G (Ia3d)nverse cubic phase of (18:1c�9/0:0)MG deduced from the water dependence.

T (◦C) Ap (nm2) vp/vl Ref.a

25 0.361 ± 0.007 0.582 ± 0.062 10.370 0.56 2,3

30 0.363 ± 0.013 0.551 ± 0.108 450 0.319 ± 0.010 0.833 ± 0.036 470 0.322 ± 0.007 0.905 ± 0.019 4

ote: A constant lipid volume of vl = 0.6285 nm3 was assumed from the density of18:1c�9/0:0)MG at 20 ◦C.

a References: 1. Chung and Caffrey (1994); 2. Pisani et al. (2001); 3. Mariani et al.1988); 4. Briggs et al. (1996).

0.721 ± 0.038 0.482 0.562 ± 0.068

nal phase of this lipid mixture, for all three different inverted cubicphases. However, the volume ratio, vp/vl , is not determined withgreat precision for the cubic phases.

Also included in Table 6 are data for the QII230/G gyroid

cubic phase of a monooleoylglycerol/dioleoyl phosphatidyl-choline/dioleoyl phosphatidylethanolamine 58:38:4 mol/mol/molternary mixture (Templer et al., 1998a). The mean pivotal area andvolume, per two chains, are close to the weighted average of thesevalues for the individual components, as demonstrated directly byTempler et al. (1998a). At the same temperature, the mean piv-otal volume per two chains is close to that for the HII-phase of(18:1c�9)2PE, which has a similar chain composition. Thus thepivotal plane for the mixture is also expected to be located approx-imately between the C4 and C5 atoms of the chains (cf. above forHII-phases).

4.3. Glycosyl dialkylglycerols

Fig. 7 shows the dependence on water content, nw, of the cubiclattice constant for the QII

230/G (Ia3d) inverse cubic phase of theglycolipid monoglucosyl didodecylglycerol (rac-(O-12:0)2Glc�DG)at 45 ◦C (Turner et al., 1992). The dependence is adequately fit byEq. (11), yielding optimized parameters: vp/vl = 0.805 ± 0.042 andAp/vl = 0.568 ± 0.016 nm−1 for the pivotal surface at 45 ◦C. With amolecular volume of vl = 0.980 nm3 (Turner et al., 1992), this givesan area per lipid at the pivotal plane of: Ap = 0.556 ± 0.016 nm2.Table 7 lists the dimensions of the gyroid inverse cubic phase ofrac-(O-12:0)2Glc�DG as a function of temperature. The pivotal vol-

4 6 8 10 12 14 16

9

10

11

12

13

cub

ic la

ttic

e c

on

sta

nt,

a (

nm

water/lipid, nw (mol/mol)

Q230/GII

, Ia3d

Fig. 7. Dependence of the cubic lattice parameter, a, on water/lipid ratio, nw , for theQII

230/G (Ia3d) gyroid inverse cubic phase of rac-didodecyl-�-d-glucosylglycerol at45 ◦C. Data from Turner et al. (1992). Solid line is a nonlinear, least-squares fit of Eq.(11) (with Eq. (12)) for the pivotal plane relative to the IPMS.

D. Marsh / Chemistry and Physics of Lipids 164 (2011) 177–183 183

Table 6Dimensions of the pivotal plane and lipid–water interface for different inverse cubic phases of dilauroyl phosphatidylcholine:lauric acid 1:2 mol/mol mixture, and of anoleoylglycerol:dioleoyl phosphatidylcholine:dioleoyl phosphatidylethanolamine mixture, deduced from the water dependence.

Lipid Phase T (◦C) Ap (nm2)a vp/vl vl (nm3)a Ref.b

(12:0)2PC/(12:0)FA 1:2 mol/molc QII230/G (Ia3d) 35–45 0.538 ± 0.018 0.855 ± 0.107 0.8675 1

QII229/P (Im3m) 35–45 0.551 ± 0.053 – 0.8675 1

QII224/D (Pn3m) 35–45 0.527 ± 0.057 0.889 ± 0.352 0.8675 1

(18:1c�9)MG/(18:1c�9)2PC/(18:1c�9)2PE 58:38:4 mol/mol QII230/G (Ia3d) 25 0.595 ± 0.011 0.748 ± 0.092 1.0190 2

a Mean value per two chains.b References: 1. Templer et al. (1998b); 2. Templer et al. (1998a).c The fatty acid is protonated.

Table 7Dimensions of the pivotal plane for the QII

230/G (Ia3d) inverse cubic phase of �-d-glucosyl didodecyl-rac-glycerol, deduced from the water dependencea.

T (◦C) Ap (nm2) vp/vl vl (nm3)

45 0.556 ± 0.016 0.805 ± 0.042 0.98050 0.549 ± 0.013 0.801 ± 0.097 0.984

5

aitcpiiei(fl

df(oapplt

sfc

A

N

R

A

B

C

D

ter, R., Erbes, J., 1998b. Inverse bicontinuous cubic phases in 2:1 fattyacid/phosphatidylcholine mixtures. The effects of chain length, hydration and

55 0.570 ± 0.063 0.771 ± 0.156 0.98760 0.589 ± 0.035 0.744 ± 0.095 0.991

a Data from Turner et al. (1992).

. Conclusion

Whereas the results for the inverse cubic phases (Figs. 6 and 7nd Tables 5–7) support a similar interpretation of the pivotal planen terms of bending elasticity as that in inverse hexagonal phases,he parameters of the pivotal plane are determined with better pre-ision from the HII-phases than from the QII-phases. In all cases, theivotal plane in the inverse phase is situated below the polar–apolar

nterface, in the hydrocarbon chain region of the lipid (i.e., vp < vl

n all cases). Depending on the lipid system, the pivotal plane isstimated to lie in the C1 to C4–C5 region of the lipid chains. Judg-ng from the profile of segmental order parameters in HII-phasesSankaram and Marsh, 1989), this corresponds to the region of leastexibility and closest packing of the lipid chains.

The area per two-chain lipid molecule at the pivotal surfaceepends on the lipid system, but lies more or less in the rangeound for the fluid lamellar (L�) phase of bilayer-forming lipidscf. Marsh, 1990). For (18:1c�9)2PE, the molecular area at the piv-tal surface is almost independent of temperature, when the chainsre flexible and packing restraints are no longer dominant. For thehosphatidylcholine:fatty acid 1:2 mixtures, the lipid area at theivotal surface is essentially independent of lipid chain length, at

east when the measurement temperature is adjusted relative tohat of chain melting.

These results define broad expectations with respect to pivotalurfaces in lipid systems and emphasise that it is the pivotal sur-ace, and not the lipid–water interface, that is relevant in definingurvature elasticity and spontaneous curvature of lipid systems.

cknowledgment

I gratefully acknowledge Christian Griesinger and the Dept. forMR-based structural biology for financial support.

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